Visualization (Exploring variation)

Motivation

Introduction to the next two lectures

Most of our visualization lectures are based on the University of Washington textbook, but the textbook doesn’t have enough material on exploratory data analysis. We therefore are supplementing with the Data Visualization and Exploratory Data Analysis material in the R for Data Science textbook (with the code translated to Altair).

• diamonds is from “Exploratory Data Analysis”
• movies is from the UW textbook
• penguins is from “Data Visualization”

What is exploratory data analysis?

Data visualization has two distinct goals

1. exploration for you to learn as much as possible
2. production for you to teach someone else what you think the key lessons are

How do the modes differ?

• When you are in exploration mode, you will look at lots of patterns and your brain filters out the noise
• Production mode is like putting a cone on your dog. You are deliberately limiting the reader’s field of vision such that they see the key messages from the plot and avoid too many distractions

The next two lectures are almost entirely about exploration. Then, in lecture 6, we will talk more about making graphics for production.

Caveat: these modes make the most sense when thinking about static visualization. Later on in the course, when we talk about dashboards, this is closer to making interfaces to help readers who don’t code explore the data.

Categorical variables

Categorical variables: roadmap

• introduce diamonds
• show table
• show bar graph

introduce dataset diamonds

from plotnine import *
from plotnine.data import diamonds, mpg
diamonds

	carat	cut	color	clarity	depth	table	price	x	y	z
0	0.23	Ideal	E	SI2	61.5	55.0	326	3.95	3.98	2.43
1	0.21	Premium	E	SI1	59.8	61.0	326	3.89	3.84	2.31
2	0.23	Good	E	VS1	56.9	65.0	327	4.05	4.07	2.31
3	0.29	Premium	I	VS2	62.4	58.0	334	4.20	4.23	2.63
4	0.31	Good	J	SI2	63.3	58.0	335	4.34	4.35	2.75
...	...	...	...	...	...	...	...	...	...	...
53935	0.72	Ideal	D	SI1	60.8	57.0	2757	5.75	5.76	3.50
53936	0.72	Good	D	SI1	63.1	55.0	2757	5.69	5.75	3.61
53937	0.70	Very Good	D	SI1	62.8	60.0	2757	5.66	5.68	3.56
53938	0.86	Premium	H	SI2	61.0	58.0	2757	6.15	6.12	3.74
53939	0.75	Ideal	D	SI2	62.2	55.0	2757	5.83	5.87	3.64

53940 rows × 10 columns

diamonds

diamonds_cut = diamonds.groupby('cut').size()
diamonds_cut

cut
Fair          1610
Good          4906
Very Good    12082
Premium      13791
Ideal        21551
dtype: int64

Categorical variables

diamonds_cut = diamonds_cut.reset_index().rename(columns={0:'N'}) # Prepare to plot

alt.Chart(diamonds_cut).mark_bar().encode(
    alt.X('cut'),
    alt.Y('N')
)

Categorical variables – summary

• this section is very brief because there’s basically only one good way to plot categorical variables with a small number of categories and this is it.
  • You can use mark_point() instead of mark_bar(), but overall, there’s a clear right answer about how to do this.
• We include this material mainly to foreshadow the fact that we will do a lot on categorical variables in the next lecture when we get to “Exploring Co-variation”

Continuous variables

Roadmap: Continuous variables

• histograms using movies
• histograms and density plots using penguins
• diamond size (carat)

Remark: The skills are absolutely fundamental and so we will intentionally be a bit repetitive.

movies dataset

movies_url = 'https://cdn.jsdelivr.net/npm/vega-datasets@1/data/movies.json'

movies = pd.read_json(movies_url)

recap scatter plot from lecture 3

alt.Chart(movies_url).mark_circle().encode(
    alt.X('Rotten_Tomatoes_Rating:Q', bin=alt.BinParams(maxbins=20)),
    alt.Y('IMDB_Rating:Q')
)

scatter plot – N movies per bin

alt.Chart(movies_url).mark_circle().encode(
    alt.X('Rotten_Tomatoes_Rating:Q', bin=alt.BinParams(maxbins=20)),
    alt.Y('count(IMDB_Rating):Q')
)

scatter plot – syntax trick

Replace count(IMDB_Rating) with count() because we aren’t using IMDB rating any more.

alt.Chart(movies_url).mark_circle().encode(
    alt.X('Rotten_Tomatoes_Rating:Q', bin=alt.BinParams(maxbins=20)),
    alt.Y('count():Q')
)

histogram using mark_bar()

hist_rt = alt.Chart(movies_url).mark_bar().encode(
    alt.X('Rotten_Tomatoes_Rating:Q', bin=alt.BinParams(maxbins=20)),
    alt.Y('count():Q')
)
hist_rt

Discussion question: how would you describe the distribution of rotten tomatoes ratings?

histogram of IMDB ratings

hist_imdb = alt.Chart(movies_url).mark_bar().encode(
    alt.X('IMDB_Rating:Q', bin=alt.BinParams(maxbins=20)),
    alt.Y('count():Q')
)
hist_imdb

Side-by-side

Discussion question – compare the two ratings distributions. which is more informative?

hist_rt | hist_imdb

introducing the penguins

from palmerpenguins import load_penguins
penguins = load_penguins()
display(penguins)

	species	island	bill_length_mm	bill_depth_mm	flipper_length_mm	body_mass_g	sex	year
0	Adelie	Torgersen	39.1	18.7	181.0	3750.0	male	2007
1	Adelie	Torgersen	39.5	17.4	186.0	3800.0	female	2007
2	Adelie	Torgersen	40.3	18.0	195.0	3250.0	female	2007
3	Adelie	Torgersen	NaN	NaN	NaN	NaN	NaN	2007
4	Adelie	Torgersen	36.7	19.3	193.0	3450.0	female	2007
...	...	...	...	...	...	...	...	...
339	Chinstrap	Dream	55.8	19.8	207.0	4000.0	male	2009
340	Chinstrap	Dream	43.5	18.1	202.0	3400.0	female	2009
341	Chinstrap	Dream	49.6	18.2	193.0	3775.0	male	2009
342	Chinstrap	Dream	50.8	19.0	210.0	4100.0	male	2009
343	Chinstrap	Dream	50.2	18.7	198.0	3775.0	female	2009

344 rows × 8 columns

histogram with steps of 200

alt.Chart(penguins).mark_bar().encode(
    alt.X('body_mass_g', bin=alt.BinParams(step=200)),
    alt.Y('count()')
)

histogram step parameter

20 vs 200 vs 200

Discussion q – what message comes from each binwidth choice?

numeric variable: transform_density()

alt.Chart(penguins).transform_density(
    'body_mass_g',
    as_=['body_mass_g', 'density']
).mark_area().encode(
    x='body_mass_g:Q',
    y='density:Q'
)

Back to diamonds, focus on carat

alt.data_transformers.disable_max_rows() # Needed because len(df) > 5000

alt.Chart(diamonds).mark_bar().encode(
    alt.X('carat', bin=alt.Bin(maxbins=10)),
    alt.Y('count()')
)

Continuous Variables

diamonds['bins'] = pd.cut(diamonds['carat'], bins=10)
diamonds.groupby('bins').size()

bins
(0.195, 0.681]    25155
(0.681, 1.162]    18626
(1.162, 1.643]     7129
(1.643, 2.124]     2349
(2.124, 2.605]      614
(2.605, 3.086]       53
(3.086, 3.567]        6
(3.567, 4.048]        5
(4.048, 4.529]        2
(4.529, 5.01]         1
dtype: int64

Continuous Variables: Typical Values

diamonds = diamonds.drop('bins', axis=1) # 'Interval' type causes plotting issues 
diamonds_small = diamonds.loc[diamonds['carat'] < 2.1] # Subset to small diamonds

alt.Chart(diamonds_small).mark_bar().encode(
    alt.X('carat', bin=alt.BinParams(step=0.1)),
    alt.Y('count()')
)

Continuous Variables: Typical Values

alt.Chart(diamonds_small).mark_bar().encode(
    alt.X('carat', bin=alt.BinParams(step=0.01)),
    alt.Y('count()')
)

Discussion questions

1. What lessons does this plot teach?
2. What questions does it raise?

Aside: “A Sunday on La Grande Jatte” by Seurat

Aside: “A Sunday on La Grande Jatte” by Seurat

Unusual numeric values (diamonds)

roadmap

• case study 1: y dimension in diamonds
  • explore some unusual values
  • three options for handling unusual values
• case study 2 (next section): cars’ gas mileage

Diamonds: examine unusual values

diamonds['y'].describe()

count    53940.000000
mean         5.734526
std          1.142135
min          0.000000
25%          4.720000
50%          5.710000
75%          6.540000
max         58.900000
Name: y, dtype: float64

Diamonds: examine unusual values

diamonds.loc[(diamonds['y'] < 3) | (diamonds['y'] > 20)] 

	carat	cut	color	clarity	depth	table	price	x	y	z
11963	1.00	Very Good	H	VS2	63.3	53.0	5139	0.00	0.0	0.00
15951	1.14	Fair	G	VS1	57.5	67.0	6381	0.00	0.0	0.00
24067	2.00	Premium	H	SI2	58.9	57.0	12210	8.09	58.9	8.06
24520	1.56	Ideal	G	VS2	62.2	54.0	12800	0.00	0.0	0.00
26243	1.20	Premium	D	VVS1	62.1	59.0	15686	0.00	0.0	0.00
27429	2.25	Premium	H	SI2	62.8	59.0	18034	0.00	0.0	0.00
49189	0.51	Ideal	E	VS1	61.8	55.0	2075	5.15	31.8	5.12
49556	0.71	Good	F	SI2	64.1	60.0	2130	0.00	0.0	0.00
49557	0.71	Good	F	SI2	64.1	60.0	2130	0.00	0.0	0.00

Diamonds: sanity check by comparing to 10 random diamonds

diamonds.sample(n=10)

	carat	cut	color	clarity	depth	table	price	x	y	z
17152	1.23	Very Good	H	VS2	62.4	58.0	6848	6.80	6.85	4.26
8388	0.35	Very Good	F	SI1	61.7	58.0	583	4.51	4.54	2.79
35711	0.30	Premium	D	VS2	60.9	58.0	911	4.35	4.32	2.64
24475	2.05	Ideal	F	SI2	62.2	57.0	12743	8.19	8.12	5.07
21128	1.03	Ideal	F	VVS2	62.4	55.6	9290	6.44	6.49	4.02
23969	2.15	Very Good	I	SI2	63.4	56.0	12100	8.20	8.15	5.18
32462	0.36	Ideal	H	VVS1	61.5	56.0	794	4.57	4.60	2.82
33301	0.51	Premium	E	I1	58.0	60.0	826	5.29	5.23	3.05
46328	0.51	Ideal	E	VS1	61.5	56.0	1758	5.11	5.16	3.16
38155	0.32	Ideal	D	VVS2	62.2	56.0	1014	4.41	4.37	2.73

What to do with unusual values?

1. Drop row
2. Code value to NA
3. Winsorize value

Diamonds: option 1 for unusual values: drop

diamonds_clean = diamonds.loc[(diamonds['y'] >= 3) | (diamonds['y'] <= 20)] 
diamonds_clean

	carat	cut	color	clarity	depth	table	price	x	y	z
0	0.23	Ideal	E	SI2	61.5	55.0	326	3.95	3.98	2.43
1	0.21	Premium	E	SI1	59.8	61.0	326	3.89	3.84	2.31
2	0.23	Good	E	VS1	56.9	65.0	327	4.05	4.07	2.31
3	0.29	Premium	I	VS2	62.4	58.0	334	4.20	4.23	2.63
4	0.31	Good	J	SI2	63.3	58.0	335	4.34	4.35	2.75
...	...	...	...	...	...	...	...	...	...	...
53935	0.72	Ideal	D	SI1	60.8	57.0	2757	5.75	5.76	3.50
53936	0.72	Good	D	SI1	63.1	55.0	2757	5.69	5.75	3.61
53937	0.70	Very Good	D	SI1	62.8	60.0	2757	5.66	5.68	3.56
53938	0.86	Premium	H	SI2	61.0	58.0	2757	6.15	6.12	3.74
53939	0.75	Ideal	D	SI2	62.2	55.0	2757	5.83	5.87	3.64

53940 rows × 10 columns

Diamonds: option 2 for unusual values: missing

diamonds['y'] = np.where((diamonds['y'] < 3) | (diamonds['y'] > 20), np.nan, diamonds['y'])
diamonds_clean = diamonds.dropna()
diamonds_clean

	carat	cut	color	clarity	depth	table	price	x	y	z
0	0.23	Ideal	E	SI2	61.5	55.0	326	3.95	3.98	2.43
1	0.21	Premium	E	SI1	59.8	61.0	326	3.89	3.84	2.31
2	0.23	Good	E	VS1	56.9	65.0	327	4.05	4.07	2.31
3	0.29	Premium	I	VS2	62.4	58.0	334	4.20	4.23	2.63
4	0.31	Good	J	SI2	63.3	58.0	335	4.34	4.35	2.75
...	...	...	...	...	...	...	...	...	...	...
53935	0.72	Ideal	D	SI1	60.8	57.0	2757	5.75	5.76	3.50
53936	0.72	Good	D	SI1	63.1	55.0	2757	5.69	5.75	3.61
53937	0.70	Very Good	D	SI1	62.8	60.0	2757	5.66	5.68	3.56
53938	0.86	Premium	H	SI2	61.0	58.0	2757	6.15	6.12	3.74
53939	0.75	Ideal	D	SI2	62.2	55.0	2757	5.83	5.87	3.64

53931 rows × 10 columns

Diamonds: option 3 for unusual values: winsorize

pctile01 = diamonds['y'].quantile(0.01)
pctile99 = diamonds['y'].quantile(0.99)

print(f"1st Percentile: {pctile01}")
print(f"99th Percentile: {pctile99}")

1st Percentile: 4.04
99th Percentile: 8.34

Diamonds: option 3 for unusual values: winsorize

diamonds['y_winsor'] = np.where(diamonds['y'] < pctile01, pctile01, 
                                np.where(diamonds['y'] > pctile99, pctile99, diamonds['y']))
diamonds

	carat	cut	color	clarity	depth	table	price	x	y	z	y_winsor
0	0.23	Ideal	E	SI2	61.5	55.0	326	3.95	3.98	2.43	4.04
1	0.21	Premium	E	SI1	59.8	61.0	326	3.89	3.84	2.31	4.04
2	0.23	Good	E	VS1	56.9	65.0	327	4.05	4.07	2.31	4.07
3	0.29	Premium	I	VS2	62.4	58.0	334	4.20	4.23	2.63	4.23
4	0.31	Good	J	SI2	63.3	58.0	335	4.34	4.35	2.75	4.35
...	...	...	...	...	...	...	...	...	...	...	...
53935	0.72	Ideal	D	SI1	60.8	57.0	2757	5.75	5.76	3.50	5.76
53936	0.72	Good	D	SI1	63.1	55.0	2757	5.69	5.75	3.61	5.75
53937	0.70	Very Good	D	SI1	62.8	60.0	2757	5.66	5.68	3.56	5.68
53938	0.86	Premium	H	SI2	61.0	58.0	2757	6.15	6.12	3.74	6.12
53939	0.75	Ideal	D	SI2	62.2	55.0	2757	5.83	5.87	3.64	5.87

53940 rows × 11 columns

When is this useful? Income data, test scores, stock returns. Important when you are using procedures where the estimates are sensitive to outliers like computing a mean or running a regression

how do I know which option to choose?

• make an educated guess by looking at the data as many ways as possible
• you often can ask your data provider… but they will quickly grow impatient so try to answer as many questions as possible yourself

Diamonds: what do we know about diamonds with outlier y values?

• depths look plausible, perhaps a bit in the tails
• carats are high but plausible
• prices are high, this would be a red flag but for the fact that carats are high too

Diamonds: what should we actually do?

My take (there is often not a ``right’’ answer or you won’t know the answer without talking to a data provider)

• Rows where x, y, and z are all zero: set to NA
• Rows where y > 20: winsorize? (hard to know for sure…)

Summary: handling unusual numeric values

Problem	Action
Erroneous row	drop row
Erroneous cell	set to NA or winsorize

How do I decide which problem I have? Examine unusual values in context of other columns (same row) and other rows (same columns). We will see this again in a future lecture.

How do I decide whether to set to NA or winsorize? Ideally, ask your data provider what’s going on with these values.

Unusual values case study

Introducing the mpg dataset

mpg

	manufacturer	model	displ	year	cyl	trans	drv	cty	hwy	fl	class
0	audi	a4	1.8	1999	4	auto(l5)	f	18	29	p	compact
1	audi	a4	1.8	1999	4	manual(m5)	f	21	29	p	compact
2	audi	a4	2.0	2008	4	manual(m6)	f	20	31	p	compact
3	audi	a4	2.0	2008	4	auto(av)	f	21	30	p	compact
4	audi	a4	2.8	1999	6	auto(l5)	f	16	26	p	compact
...	...	...	...	...	...	...	...	...	...	...	...
229	volkswagen	passat	2.0	2008	4	auto(s6)	f	19	28	p	midsize
230	volkswagen	passat	2.0	2008	4	manual(m6)	f	21	29	p	midsize
231	volkswagen	passat	2.8	1999	6	auto(l5)	f	16	26	p	midsize
232	volkswagen	passat	2.8	1999	6	manual(m5)	f	18	26	p	midsize
233	volkswagen	passat	3.6	2008	6	auto(s6)	f	17	26	p	midsize

234 rows × 11 columns

Q: Why do some cars have better than typical mileage?

potential_outliers = mpg.loc[(mpg["hwy"] > 40) | ((mpg["hwy"] > 20) & (mpg["displ"] > 5))]
potential_outliers

	manufacturer	model	displ	year	cyl	trans	drv	cty	hwy	fl	class
23	chevrolet	corvette	5.7	1999	8	manual(m6)	r	16	26	p	2seater
24	chevrolet	corvette	5.7	1999	8	auto(l4)	r	15	23	p	2seater
25	chevrolet	corvette	6.2	2008	8	manual(m6)	r	16	26	p	2seater
26	chevrolet	corvette	6.2	2008	8	auto(s6)	r	15	25	p	2seater
27	chevrolet	corvette	7.0	2008	8	manual(m6)	r	15	24	p	2seater
158	pontiac	grand prix	5.3	2008	8	auto(s4)	f	16	25	p	midsize
212	volkswagen	jetta	1.9	1999	4	manual(m5)	f	33	44	d	compact
221	volkswagen	new beetle	1.9	1999	4	manual(m5)	f	35	44	d	subcompact
222	volkswagen	new beetle	1.9	1999	4	auto(l4)	f	29	41	d	subcompact

Note: calling geom_point() more than once!

Q: Why do some cars have better than typical mileage?

base = alt.Chart(mpg).mark_point().encode(
         alt.X('displ:Q', title = "Engine size (displ)"),
         alt.Y('hwy:Q', title = "Gas mileage")
    ).properties(
        width=600, 
        height=400 )

outliers = alt.Chart(potential_outliers).mark_point(
    color='red',
    size=100,
    shape='circle'
    ).encode(
        x='displ:Q',
        y='hwy:Q'
    ).properties(
        width=600, 
        height=400)
plot = base + outliers
plot

Q: Why do some cars have better than typical mileage?

labels = alt.Chart(potential_outliers).mark_text(
    align='left',
    dx=10,  # Adjust horizontal distance of text from the point
    dy=-5   # Adjust vertical distance of text from the point
).encode(
    alt.X('displ:Q', title = "Engine size (displ)"),
    alt.Y('hwy:Q', title = "Gas mileage"),
    text='model:N'  # Display car_model as the label
).properties(
    width=600, 
    height=400)

plot = base + outliers + labels
plot

Q: How are there big engines and good mileage? color

alt.Chart(mpg).mark_point(size=100).encode(
    x='displ:Q',  # Quantitative variable for displacement
    y='hwy:Q',    # Quantitative variable for highway mpg
    color='class:N',  # Categorical variable for class
    tooltip=['displ', 'hwy', 'class']  # Optional: tooltip to display values on hover
)

gas mileage summary

• Question: Why do some cars have better than typical mileage? (What’s going on with these outliers?)
  • Tools:
    • identify outliers
    • color = class
      
  • Answer: 2-seaters & subcompact